Violas Group Presentation V1
Joe, Jason, Pam
August 4, 2018
Review the Guns dataset from the “AER”" package.
- Dataset is a balanced panel of data on 50 US states, plus the District of Columbia (for a total of 51 states), by year for 1977?????1999, and contains 1,173 observations on 13 variables. The variables are:
- state: factor indicating state.
- year: factor indicating year.
- violent: violent crime rate (incidents per 100,000 members of the population).
- murder: murder rate (incidents per 100,000).
- robbery: robbery rate (incidents per 100,000).
- prisoners: incarceration rate in the state in the previous year (sentenced prisoners per 100,000 residents;value for the previous year).
- afam: percent of state population that is African-American, ages 10 to 64.
- cauc: percent of state population that is Caucasian, ages 10 to 64.
- male: percent of state population that is male, ages 10 to 29.
- population: state population, in millions of people.
- income: real per capita personal income in the state (US dollars).
- density: population per square mile of land area, divided by 1,000.
- law: factor. Does the state have a shall carry law in effect in that year?
Objective: Using the data collected see if we can Identify a model that helps best predict the violence rate.
Review a structure of the data
## 'data.frame': 1173 obs. of 13 variables:
## $ year : Factor w/ 23 levels "1977","1978",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ violent : num 414 419 413 448 470 ...
## $ murder : num 14.2 13.3 13.2 13.2 11.9 10.6 9.2 9.4 9.8 10.1 ...
## $ robbery : num 96.8 99.1 109.5 132.1 126.5 ...
## $ prisoners : int 83 94 144 141 149 183 215 243 256 267 ...
## $ afam : num 8.38 8.35 8.33 8.41 8.48 ...
## $ cauc : num 55.1 55.1 55.1 54.9 54.9 ...
## $ male : num 18.2 18 17.8 17.7 17.7 ...
## $ population: num 3.78 3.83 3.87 3.9 3.92 ...
## $ income : num 9563 9932 9877 9541 9548 ...
## $ density : num 0.0746 0.0756 0.0762 0.0768 0.0772 ...
## $ state : Factor w/ 51 levels "Alabama","Alaska",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ law : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...Review data visually
The violent variable does not appear to be normally distributed
- Histogram, the distribution of the violent variable deviates from the normal distribution
- Doesn’t adhere to the blue QQ-line on the the QQ-Plot.
- Violent variable is skewed to the right (positively) and the mean (503.074) > median (443)
- Sort Plot there appears to be a few clusters on the right side of the plot, where the violent crime rate is high
- Consider the use a ?log()? transformation on this variable possibly reduce the skewness
- Potentially make it easier to interpret whether or not a relationship exist between variables that might not have otherwise been obvious.
Scatterplot - Best way to show this?
## Loading required package: ggplot2
* The variables murder and robbery have the strongest positive linear relationship with the violent crime rate, with a correlation value of 0.827 and 0.907 + Considering these variables are components of the violent crime rate, we expected there to be high correlation values + These variables are more of a subcategory of violent crimes and therefore aren’t good predictors for a linear model * Other variables that showed a high correlation value were prisoners (0.703), density (0.665), afam (0.57), cauc(-0.573), and income (0.408) + The law variable was also a good indicator because the box plot showed that on average states that have a shall carry law in effect have a lower violent crime rate as well as as a lower murder and robbery rate.
Fit Plot
* The Fit Plot shows that the actual values fluctuate fairly significantly from the fitted values(pink reference line) * The Residual plot shows that a majority of the observed values fall within +/-500 of the fitted values + This could be considered a substantial deviation from the fitted values + Next lets examine if a log transformation or box-cox transformation would be useful
Log Transformation Fitted Plot & Transformation Residuals vs. Fitted
* The Log Transformed Fit Plot shows that the actual values fluctuate fairly significantly from the fitted values(blue reference line). * The Transformed Fit Plot vs Residual Plot is easier to read than the previous model (m5) + Y-axis, which shows the .resid is shown in Z-scores with a majority of the data falling between +/- 1 standard deviations from the mean + Moving forward we will use the log transformation of the violent variable to make it easier to interpret the data
****Check on the m5 statement******
Box-Cox Transformation
## Estimated transformation parameter
## Y1
## 0.6022324Should we remove this line from showing in the html?
Box-Cox Transformation Fitted Plot & Transformation Residuals vs. Fitted
## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'
Normal QQ-plots for detecting non normality
Although there is not much seperation the Box-Cox Tansformed model seems to fit the best
Histogram
*The residuals of the Box-Cox Transformed model from the above plots appears to have the most normally distributed histogram
Shapiro test of Normality
##
## Shapiro-Wilk normality test
##
## data: residuals(m1)
## W = 0.97023, p-value = 8.514e-15##
## Shapiro-Wilk normality test
##
## data: residuals(m2)
## W = 0.97727, p-value = 1.293e-12##
## Shapiro-Wilk normality test
##
## data: residuals(mlam)
## W = 0.98664, p-value = 6.98e-09- The Shapiro-Wilk test indicates the Pearson correlation between the residuals and the normal quantiles, W is equal to:
- 0.97023 untransformed model
- 097727 log-transformed model
- 0.98664 box-cox transformed model
Based on our analysis we will utilize the box-cox transformed model
Stepwise Regression
## Start: AIC=2239.81
## violent^lam ~ year + murder + robbery + prisoners + afam + cauc +
## male + population + income + density + state + law
##
## Df Sum of Sq RSS AIC
## - cauc 1 0 6872 2237.8
## - density 1 0 6873 2237.9
## - afam 1 6 6879 2238.9
## - population 1 7 6879 2239.0
## - income 1 8 6880 2239.2
## <none> 6872 2239.8
## - law 1 78 6950 2251.0
## - male 1 100 6972 2254.7
## - murder 1 132 7005 2260.2
## - prisoners 1 218 7090 2274.4
## - year 22 1632 8504 2445.7
## - robbery 1 4594 11467 2838.3
## - state 50 33830 40702 4226.3
##
## Step: AIC=2237.81
## violent^lam ~ year + murder + robbery + prisoners + afam + male +
## population + income + density + state + law
##
## Df Sum of Sq RSS AIC
## - density 1 0 6873 2235.9
## - population 1 8 6880 2237.1
## - income 1 8 6881 2237.2
## <none> 6872 2237.8
## - afam 1 16 6888 2238.5
## - law 1 78 6951 2249.1
## - murder 1 133 7006 2258.3
## - male 1 194 7066 2268.4
## - prisoners 1 231 7104 2274.6
## - year 22 1825 8697 2470.0
## - robbery 1 4595 11467 2836.4
## - state 50 33910 40782 4226.6
##
## Step: AIC=2235.88
## violent^lam ~ year + murder + robbery + prisoners + afam + male +
## population + income + state + law
##
## Df Sum of Sq RSS AIC
## - population 1 7 6880 2235.1
## - income 1 9 6882 2235.4
## <none> 6873 2235.9
## - afam 1 23 6896 2237.8
## - law 1 79 6952 2247.3
## - murder 1 136 7009 2256.9
## - male 1 193 7066 2266.4
## - prisoners 1 390 7263 2298.6
## - year 22 1825 8698 2468.1
## - robbery 1 4594 11467 2834.4
## - state 50 43238 50111 4466.2
##
## Step: AIC=2235.13
## violent^lam ~ year + murder + robbery + prisoners + afam + male +
## income + state + law
##
## Df Sum of Sq RSS AIC
## - income 1 9 6889 2234.6
## <none> 6880 2235.1
## - afam 1 18 6898 2236.2
## - law 1 78 6958 2246.4
## - murder 1 130 7010 2255.0
## - male 1 199 7080 2266.6
## - prisoners 1 452 7332 2307.8
## - year 22 1849 8729 2470.3
## - robbery 1 4637 11517 2837.4
## - state 50 47934 54814 4569.5
##
## Step: AIC=2234.63
## violent^lam ~ year + murder + robbery + prisoners + afam + male +
## state + law
##
## Df Sum of Sq RSS AIC
## <none> 6889 2234.6
## - afam 1 15 6904 2235.1
## - law 1 89 6978 2247.7
## - murder 1 172 7061 2261.5
## - male 1 206 7095 2267.2
## - prisoners 1 455 7344 2307.7
## - year 22 1844 8733 2468.9
## - robbery 1 4688 11577 2841.6
## - state 50 48242 55131 4574.2From the stepwise regression we see that cauc, density, and income both should be removed from the data to create a globally optimal model
Compare Coefficients of the two models
## Calls:
## 1: lm(formula = violent^lam ~ ., data = Guns)
## 2: lm(formula = violent^lam ~ year + murder + robbery + prisoners +
## afam + male + state + law, data = Guns)
##
## Model 1 Model 2
## (Intercept) 11.3 11.6
## year1978 0.886 0.955
## year1979 2.32 2.39
## year1980 2.65 2.67
## year1981 2.41 2.45
## year1982 2.57 2.60
## year1983 2.32 2.39
## year1984 3.26 3.42
## year1985 4.14 4.35
## year1986 5.33 5.58
## year1987 5.46 5.75
## year1988 6.41 6.74
## year1989 6.99 7.36
## year1990 8.93 9.30
## year1991 9.55 9.90
## year1992 10.3 10.7
## year1993 10.7 11.1
## year1994 10.3 10.8
## year1995 9.96 10.41
## year1996 8.94 9.42
## year1997 8.90 9.43
## year1998 8.08 8.69
## year1999 6.98 7.63
## murder 0.152 0.161
## robbery 0.0499 0.0496
## prisoners 0.00993 0.01060
## afam -0.419 -0.337
## cauc -0.00345
## male 1.12 1.15
## population 0.148
## income 0.000135
## density -0.354
## stateAlaska 1.21 1.56
## stateArizona -0.53089 -0.00311
## stateArkansas -3.57 -3.56
## stateCalifornia -0.905 3.402
## stateColorado -4.31 -3.37
## stateConnecticut -9.31 -8.10
## stateDelaware -0.431 -0.277
## stateDistrict of Columbia 3.678 -0.974
## stateFlorida 9.82 11.71
## stateGeorgia -2.05 -1.61
## stateHawaii -7.67 -8.70
## stateIdaho -11.4 -11.1
## stateIllinois 1.08 3.01
## stateIndiana -7.29 -6.28
## stateIowa -12.8 -12.0
## stateKansas -7.50 -6.87
## stateKentucky -10.5 -10.1
## stateLouisiana 4.42 4.26
## stateMaine -17.1 -16.5
## stateMaryland 3.05 3.60
## stateMassachusetts 0.765 2.131
## stateMichigan -0.824 0.603
## stateMinnesota -13.5 -12.3
## stateMississippi -7.39 -8.02
## stateMissouri -2.36 -1.51
## stateMontana -15.6 -15.4
## stateNebraska -10.19 -9.57
## stateNevada -2.67 -2.23
## stateNew Hampshire -19.2 -18.3
## stateNew Jersey -5.28 -3.92
## stateNew Mexico 8.06 8.09
## stateNew York -3.505 -0.626
## stateNorth Carolina -1.45 -0.91
## stateNorth Dakota -25.0 -24.7
## stateOhio -9.26 -7.54
## stateOklahoma -3.22 -2.97
## stateOregon -3.06 -2.25
## statePennsylvania -10.26 -8.19
## stateRhode Island -7.29 -6.98
## stateSouth Carolina 10.4 10.1
## stateSouth Dakota -17.0 -16.7
## stateTennessee -1.484 -0.917
## stateTexas -5.87 -3.51
## stateUtah -13.1 -12.7
## stateVermont -18.9 -18.4
## stateVirginia -12.1 -11.3
## stateWashington -4.34 -3.20
## stateWest Virginia -15.5 -15.1
## stateWisconsin -16.9 -15.8
## stateWyoming -10.7 -10.3
## lawyes -1.09 -1.14As you can see the larger model model mlam has larger standard errors then the smaller model m3, which has gone through the stepwise regression
P-Value of the Partial F-test
## Analysis of Variance Table
##
## Model 1: violent^lam ~ year + murder + robbery + prisoners + afam + male +
## state + law
## Model 2: violent^lam ~ year + murder + robbery + prisoners + afam + cauc +
## male + population + income + density + state + law
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1094 6889.0
## 2 1090 6872.4 4 16.557 0.6565 0.6224Conclusion
The results produced by the anova function show a non-significant result (p-value = 0.6224). Therefore we should reject the larger model (mlam) and move forward with the smaller model (m3).
To be completed by JH and added potentially
Checking for any influential points that are controlling the results
## StudRes Hat CookD
## 185 0.4156464 0.25978614 0.000731069
## 189 -5.0794642 0.25204347 0.102420286
## 195 3.9512822 0.13445287 0.028833267
## 207 0.4733689 0.29029093 0.001105054
## 1127 4.8407145 0.06827768 0.020271504- As all of the Cooks values are low then no single varaible is controlling the results
Transformation Check
* After looking at the scaled scatterplot of that data we determine that cauc needs to be transformed
Stepwise regression with cauc normalized with log
## Start: AIC=9240.41
## violent ~ year + murder + robbery + prisoners + afam + log(cauc) +
## male + population + income + density + state + law
##
## Df Sum of Sq RSS AIC
## - density 1 236 2685757 9238.5
## - log(cauc) 1 335 2685857 9238.6
## <none> 2685522 9240.4
## - income 1 5124 2690646 9240.6
## - afam 1 6355 2691877 9241.2
## - male 1 18702 2704223 9246.5
## - population 1 24164 2709685 9248.9
## - law 1 26090 2711611 9249.7
## - prisoners 1 177230 2862751 9313.4
## - murder 1 332563 3018084 9375.3
## - year 22 502422 3187943 9397.6
## - robbery 1 3052545 5738066 10129.0
## - state 50 8884729 11570250 10853.6
##
## Step: AIC=9238.51
## violent ~ year + murder + robbery + prisoners + afam + log(cauc) +
## male + population + income + state + law
##
## Df Sum of Sq RSS AIC
## - log(cauc) 1 219 2685976 9236.6
## <none> 2685757 9238.5
## - income 1 5544 2691301 9238.9
## - afam 1 6479 2692237 9239.3
## - male 1 18570 2704327 9244.6
## - population 1 24029 2709786 9247.0
## - law 1 26389 2712146 9248.0
## - prisoners 1 303717 2989474 9362.2
## - murder 1 336617 3022375 9375.0
## - year 22 502244 3188001 9395.6
## - robbery 1 3058367 5744124 10128.2
## - state 50 9307137 11992894 10893.7
##
## Step: AIC=9236.6
## violent ~ year + murder + robbery + prisoners + afam + male +
## population + income + state + law
##
## Df Sum of Sq RSS AIC
## <none> 2685976 9236.6
## - income 1 5838 2691814 9237.2
## - afam 1 16775 2702751 9241.9
## - law 1 26764 2712740 9246.2
## - male 1 27232 2713208 9246.4
## - population 1 27239 2713215 9246.4
## - prisoners 1 303511 2989487 9360.2
## - murder 1 336791 3022767 9373.2
## - year 22 527892 3213868 9403.1
## - robbery 1 3136507 5822483 10142.1
## - state 50 10803349 13489325 11029.7Interestingly this leads to the stepwise regression only from removing density and cauc from the data set but leaving income.
## Analysis of Variance Table
##
## Model 1: violent^lam ~ year + murder + robbery + prisoners + afam + male +
## state + law
## Model 2: violent^lam ~ year + murder + robbery + prisoners + afam + cauc +
## male + population + income + density + state + law
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1094 6889.0
## 2 1090 6872.4 4 16.557 0.6565 0.6224Cross-Validation for Linear Models
Cross-Validation for Linear Models against 10 seeds each
## mse.m1 mse.m2 mse.m3 mse.m4 mse.mlam
## 1 2889.625 0.01761853 7.150634 2890.016 7.201197
## 2 2890.438 0.01885826 7.429532 2834.517 7.645189
## 3 2738.429 0.01737850 6.972956 2742.381 6.989694
## 4 2859.606 0.01764531 7.123702 2836.695 7.216332
## 5 2712.441 0.01768070 6.987390 2706.323 7.057505
## 6 2919.177 0.01747342 7.073046 2921.903 7.136046
## 7 2933.971 0.01829054 7.059834 2922.766 7.310023
## 8 2712.441 0.01768070 6.987390 2706.323 7.057505
## 9 2919.177 0.01747342 7.073046 2921.903 7.136046
## 10 2933.971 0.01829054 7.059834 2922.766 7.310023## Analysis of Variance Table
##
## Model 1: violent^lam ~ year + murder + robbery + prisoners + afam + male +
## state + law
## Model 2: violent^lam ~ year + murder + robbery + prisoners + afam + cauc +
## male + population + income + density + state + law
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1094 6889.0
## 2 1090 6872.4 4 16.557 0.6565 0.6224```